Evolution of the superconducting state through quantum criticality in CeRh$_{1-x}$Co$_x$In$_5$

The Ce-based 115 materials exhibit a host of novel ground states separated by experimentally tunable quantum instabilities. In the single-crystal alloy series CeRh$_{1-x}$Co$_x$In$_5$, long range antiferromagnetic order is gradually suppressed upon chemical substitution of Co for Rh and followed by a robust superconducting state extending to the 2.3~K transition of the infamous heavy-fermion superconductor CeCoIn$_5$. Here we present a thorough study of heat transport measurements of high- quality single crystals of CeRh$_{1-x}$Co$_x$In$_5$ for several different superconducting samples spanning both the coexistent magnetic and non-magnetic regions of the $x$-$T$ phase diagram. By extracting the residual ($T\to 0$ limit) electronic thermal conductivity of samples at several $x$ values, we analyze the nature of the superconducting state on either side of the incipient quantum critical point near $x\simeq 0.65$ and study the influence of coexistent magnetism on the pairing state of these materials.